×

zbMATH — the first resource for mathematics

Generalization of Runge-Kutta discontinuous Galerkin method to LWR traffic flow model with inhomogeneous road conditions. (English) Zbl 1067.65105
The paper presents an algorithm generalizing the Runge-Kutta discontinuous Galerkin finite element method applied to the scalar hyperbolic conservation equation with a spacially varying flux. The equation is known as the Lightill-Whitham-Richards (LWR) model which is considered with discontinuity terms here. The paper contributes in the redesign of a numerical flux and a limiter resulting in exactly steady flows and stationary discontinuities. The key idea for the design is that the time evolution would follow the characteristic direction. Numerical simulations involving two fluxes and two limiters are supplied.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
90B20 Traffic problems in operations research
35L65 Hyperbolic conservation laws
35R05 PDEs with low regular coefficients and/or low regular data
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lighthill, Proc Roy Soc London Ser A 229 pp 281– (1955)
[2] Richards, Operations Res 4 pp 42– (1956)
[3] Chowdhury, Phys Rep 323 pp 199– (2000)
[4] Helbing, Rev Modern Phys 73 (2001)
[5] Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.
[6] Zhang, J Comput Appl Math 156 pp 1– (2003) · Zbl 1031.35104
[7] Tveito, SIAM J Sci Comput 16 pp 320– (1995)
[8] Klausen, J Differ Eq 157 pp 41– (1999)
[9] Upwind differencing for hyperbolic systems of conservation laws, Numerical Methods for Engineering, Vol. 1, Dunod, Paris, 1980, pp 137-149.
[10] Roes, J Comp Phys 43 pp 357– (1981)
[11] Harten, J Comput Phys 49 pp 357– (1983)
[12] Harten, SIAM Rev 25 (1983)
[13] Cockburn, Math Comput 52 pp 411– (1989)
[14] Cockburn, J Comput Phys 84 pp 90– (1989)
[15] Cockburn, Math Comput 54 pp 545– (1990)
[16] Cockburn, M2AN 25 pp 337– (1991)
[17] Lecture Notes in Mathematics-An Introduction to the Discontinuous Galerkin Method for Convection-Dominated Problems [R]: 1697, Springer, Cetraro, Italy: 1997, pp. 151-185.
[18] Ru-Xun, Math Comput Simulation 56-1 pp 55– (2001)
[19] Peng, Chinese J Comput Phys 18 pp 229– (2001)
[20] Peng, Chinese J Comput Phys 19 pp 142– (2002)
[21] Peng, Chinese J Comput Phys 20 pp 130– (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.